3.9 Uniform convergence

1. Prove that if a sequence $\left\{f_n\right\}$ of functions converges uniformly on a set $S$ to a function $f$ then it converges pointwise to $f.$
2. Let $S=(0,1),$ and for each $n$ define $f_n\left(x\right)=x^n.$ Prove that $\left\{f_n\right\}$ converges pointwise to the zero function, but that $\left\{f_n\right\}$ does not converge uniformly to the zero function. Conclude that pointwise convergence does not imply uniform convergence. HINT: Suppose the sequence does converge uniformly.Take $ϵ=1/2,$ let $N$ be a corresponding integer, and consider $x$ 's of the form $x=1-h$ for tiny $h$ 's.
3. Suppose the sequence $\left\{f_n\right\}$ converges uniformly to $f$ on $S,$ and the sequence $\left\{g_n\right\}$ converges uniformly to $g$ on $S.$ Prove that the sequence $\{f_n+g_n\}$ converges uniformly to $f+g$ on $S.$
4. Suppose $\left\{f_n\right\}$ converges uniformly to $f$ on $S,$ and let $c$ be a constant. Show that $\left\{c{f}_n\right\}$ converges uniformly to $cf$ on $S.$
5. Let $S=R,$ and set $f_n\left(x\right)=x+(1/n).$ Does $\left\{f_n\right\}$ converge uniformly on $S?$ Does $\left\{f_n^2\right\}$ converge uniformly on $S?$ What does this say about the limit of a product of uniformly convergent sequences versus the product of the limits?
6. Suppose $a$ and $b$ are nonnegative real numbers and that $|a-b|<{ϵ}^2.$ Prove that $|\sqrt{a}-\sqrt{b}|<2ϵ.$ HINT: Break this into cases, the first one being when both $\sqrt{a}$ and $\sqrt{b}$ are less than $ϵ.$
7. Suppose $\left\{f_n\right\}$ is a sequence of nonnegative real-valued functions that converges uniformly to $f$ on $S.$ Use part (f) to prove that the sequence $\left\{\sqrt{f_n}\right\}$ converges uniformly to $\sqrt{f}.$
8. For each positive integer $n,$ define $f_n$ on $(-1,1)$ by $f_n\left(x\right)={\left|x\right|}^{1+1/n}.$ Prove that the sequence $\left\{f_n\right\}$ converges uniformly on $(-1,1)$ to the function $f\left(x\right)=\left|x\right|.$ HINT: Let $ϵ>0$ be given. Consider $\left|x\right|$ 's that are $<ϵ$ and $\left|x\right|$ 's that are $\ge ϵ.$ For $\left|x\right|<ϵ,$ show that $|f_n\left(x\right)-f\left(x\right)|<ϵ$ for all $n.$ For $\left|x\right|\ge ϵ,$ choose $N$ so that $|{ϵ}^{1/n}-1|<ϵ.$ How?

Let $\left\{f_n\right\}$ be a sequence of functions on a set $S,$ let $f$ be a function on $S,$ and suppose that for each $n$ we have $|f\left(x\right)-f_n\left(x\right)|<1/n$ for all $x\in S.$ Prove that the sequence $\left\{f_n\right\}$ converges uniformly to $f.$